Rheology of Variable Viscosity-Based Mixed Convective Inclined Magnetized Cross Nanofluid with Varying Thermal Conductivity

Abstract

Cross nanofluid possesses an extraordinary quality among the various fluidic models to explore the key characteristics of flowing fluid during very low and very high shear rates and its viscosity models depend upon shear rate. The current study establishes the numerical treatment regarding variable viscosity-based mixed convective inclined magnetized Cross nanofluid with varying thermal conductivities over the moving permeable surface. Along with variable thermal conductivities, we considered thermal radiation, thermophoresis, and the Brownian motion effect. An inclined magnetic field was launched for velocity scrutiny and the heat transfer fact was numerically seen by mixed convective conditions. Similarity variables were actioned on generated PDEs of the physical model and conversion was performed into ODEs. Numerical results showed that the frictional force and Nusselt quantity considerably influence the skinning heat transfer processes over the geometry of a moving permeable surface. Furthermore, less velocity was noticed for the greater suction parameter and the Brownian motion parameter corresponds to lower mass transport.

1. Introduction

Non-Newtonian fluids have a vital significance in industry, engineering, and daily life, such as in fluid friction reduction, scale-up, and flow tracers, custard, paint, blood, shampoo, molten polymers, starch suspensions, surfactant applications to large-scale heating and cooling systems, melted butter, toothpaste, corn starch, salt solutions, oil–pipeline friction reduction, lubrication and in biomedical flows. Several models [1,2,3,4,5,6,7,8] are established for checking the behavior of non-Newtonian fluids but the most attractive model, Cross fluid, has astonishing key features for investigating the fluid behavior during very high and very low shear rates. No other model can possess such ability in such a situation; due to this factor Cross nanofluid has been recognized as the most important numerical model by recent researchers [9,10,11,12,13,14,15]. Azam et al. [16] conducted research on numerical modeling for the changeable thermal characteristics and heat source (sink) in a Cross nanofluid flowing across a movable cylinder. In this analysis, it was concluded that thermal conductivity makes for higher temperatures. Furthermore, Azam et al. [17] investigated the role of activation energy in the formation of cross-links in the axially symmetric flow of a thermal radiation Cross nanofluid. The importance of bioconvection flow in the Cross nanofluid with multiple slips over the associated wedge geometry is revealed by [18]. In this study, authors utilized gyrotactic motile microorganisms. Haq et al. [19] made their investigation about bio-convection in a Cross nanofluid. in this study, the authors engaged the facts of activation energy, magnetic field, and gyrotactic microorganisms. Magneto Cross fluid associated with bio-convection, activation energy, and gyrotactic microorganisms has been studied [20]. The latest study [21] related to the saturated flow of Cross nanofluid with multiple solutions is made over the vertical thin needle point geometry. Unsteady fluid flow over the Falkner–Skan wedge with its activation energy, nonlinear thermal radiation, and melting heat process is revealed by Waqas et al. [22].

Nanofluid not only makes heat transmission possible but also reduces energy consumption. Because of its importance in biological molecules, scientists are working in several ways to improve it, such as in heat management or energy storage systems, nano cryotherapy, electric generation, heat transfer, and so on. The first-time revelation of the nanofluids concept was established by Choi et al. [23] and for the first time it is established that nanofluid-containing nanoparticles can enhance thermal conductivity. Said et al. [24] published their article on the issue of recent advances in nanofluids with exergy, solar energy, environmental impact, and economic analysis. Sheikholeslami et al. [25] did work on the progress on photovoltaic systems and flat plate solar collectors in nanofluid. A numerical study related to the magneto-convective flow considered nanoparticles of copper–water nanofluid and entropy generation with Chamfers, by Marzougui et al. [26]. The significance of nonlinear thermal radiation and activation energy in a 3D mathematical model of Eyring–Powell nanofluid is presented by Muhammad et al. [27]. Studies related to fundamental and physical stability, heat transport, thermophysical properties, dynamic motion, and applications, as well as the challenges of nanofluids, are described by Said et al. [28]. Meibodi et al. [29] analyzed the second law analysis of a nanofluid-based solar collector using experimental data. Furthermore, many other scholars [30,31] performed investigations related to the financial and environmentally friendly analysis of metallic oxide nanoliquid and the applications of nanofluids in solar energy.

There are many research articles on Cross fluid associated with several effects, but variable viscosity and variable thermal conductivity with the inclined magnetized flow are still not investigated as they should be so, the current study established the numerical treatment regarding variable viscosity based on a mixed convective inclined magnetized Cross nanofluid with varying thermal conductivity over the moving permeable surface. Along with variable thermal conductivity, we considered thermal radiation, thermophoresis, and the Brownian motion effect. An inclined magnetic field was launched for velocity scrutiny and the heat transfer fact was numerically observed by mixed convective conditions.

2. Viscosity Model of Cross Nanofluid

The Cauchy stress tensor for the case of a four-parameter fluid can be defined as

$$\tau =-pI+\mu \left(\dot{\gamma }\right)A_1$$

(1)

The shear rate term comprising of zero and infinite shear rate viscosities along with shear power-law index $n$ and material time constant $\Gamma$ is given by

$$\mu \left(\dot{\gamma }\right)=\left[{\mu }_{\infty }+\left({\mu }_o-{\mu }_{\infty }\right)\left[1+{(\Gamma \dot{\gamma })}^n\right]\right].$$

(2)

The first Rivilin–Ericksen tensor is given by

$$\left(\dot{\gamma }=\sqrt{\frac{1}{2}tr{\left(A_1\right)}^2},A_1=\left(\Delta V\right)+{\left(\Delta V\right)}^T\right)$$

(3)

3. Orientation and Formulation of Physical Problem

It is considered that the 2D Cross nanofluid is flowing, facing a stagnation point in a porous medium. Furthermore, fluid is considered to be incompressible very near to the stagnation point and the surface coincides with the plane of equation $y=0$. The zero-mass flux and mixed convective temperature conditions can be considered at the boundary surface of the sheet. The active and passive controls of nanoparticles at the surface of the sheet are controlled by zero mass flux nanoparticle conditions. The stagnation point is at the point $O$. The fluid region is taken as $y>0$, and the above sheet is porous, which can be seen in Figure 1. Continuity, momentum, temperature, and concentration equations are derived from the law of conservation of momentum, Navier Stoke’s equation, the second law of thermodynamics, and Fick’s second law of diffusion respectively. The variation in fluid, temperature and concentration are monitored with the utilization of variable viscosity [32], variable thermal conductivity [33,34], and variable molecular diffusivity phenomena, respectively. The governing system of equations in the light of the above-mentioned assumptions is given below.

$$u_x+v_y=0$$

(4)

$${\rho }_f\left[u{u}_x+v{v}_x\right]=\frac{\partial }{\partial y}\left\{\frac{{\mu }_0}{{\left[1+{\Gamma }^2\left(u_y\right)\right]}^n}\right\}+\frac{\sigma B_0^2}{\rho }{\sin }^2\left(\omega \right)\left(u-u_e\right)+u_e\frac{d{u}_e}{dx}+g{\beta }_T\left(T-T_{\infty }\right)$$

(5)

$$\left[u{T}_x+v{T}_y\right]=\frac{1}{\rho c_p}\frac{\partial }{\partial y}\left[K\left(T\right)T_y\right]+\tau \left[D_B{T}_y{C}_y+\frac{D_T}{T_{\infty }}\left[{\left(T_y\right)}^2\right]\right]+\frac{16{\sigma }^{\ast }}{3k^{\ast }\rho c_p}\left[\frac{\partial }{\partial y}\left(T_y{T}^3\right)\right]$$

(6)

$$\left[u{C}_x+v{C}_y\right]=\frac{1}{\rho c_p}\frac{\partial }{\partial y}\left[D\left(C\right)C_y\right]+\left[D_B{C}_{yy}+\frac{D_T}{T_{\infty }}\left(T_{yy}\right)\right]$$

(7)

where $\left[\begin{array}{l}K\left(T\right)=k_{\infty }\left\{1+{\epsilon }_1\frac{\left(T-T_{\infty }\right)}{\Delta T}\right\},and D\left(C\right)=D_{\infty }\left\{1+{\epsilon }_2\frac{\left(C-C_{\infty }\right)}{\Delta C}\right\}\\ \frac{1}{\mu }=\frac{1}{{\mu }_{\infty }}\left[1+{\gamma }_1\left(T-T_{\infty }\right)\right],or \frac{1}{\mu }=\alpha \left\{T-T_{\Gamma }\right\}\end{array}\right.$.

Following the boundary conditions must be considered for Equations (4)–(7).

$$\left.\begin{array}{l}{u}_w=cx,-K\left(T\right)T_y=\left(T-T_W\right),v=v_w,\left[D_B{C}_y+\frac{D_T}{T_{\infty }}\left(T_y\right)\right]=0,at y=0,\\ C\to C_{\infty },u\to u_e=ax,T\to T_{\infty },as y \to \infty \end{array}\right\}$$

(8)

The system of PDE (4)–(8) is passed through these similar transformations

$$\left.\begin{array}{l}u=xc{f}^{\prime }\left(\eta \right),\hspace{0.33em}\eta =y{\left(\frac{\upsilon }{c}\right)}^{-1/2},v=-\sqrt{cv}f\left(\eta \right)\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_0},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_0}\end{array}\right\}$$

(9)

$$\begin{gathered} f^{‴}{\left(1+{\left(w{e}_1{f}^{″}\right)}^n\right)}^{-2}\left\{\left[1-\left(n-1\right){\left(We{f}^{″}\right)}^n\right]\right\}+f^{″}\left(\frac{{\theta }^{\prime }}{{\theta }_r-\theta }\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(\frac{{\theta }_r-\theta }{{\theta }_r}\right)\left[f^{\prime 2}-f{f}^{″}+G\theta +A^2-1-M^2{\sin }^2\left(\omega \right)f^{\prime }\right]=0 \end{gathered}$$

(10)

$$\left(1+{\epsilon }_1\right){\theta }^{″}+{\epsilon }_1{\theta }^{\prime 2}+\frac{4}{3}N\frac{d}{d\eta }\left[{\left\{\theta \left({\theta }_f-1\right)+1\right\}}^3{\theta }^{\prime }\right]+\mathrm{Pr}\left[Nb{\varphi }^{\prime }{\theta }^{\prime }+N_t{\theta }^{\prime 2}+\left(f{\theta }^{\prime }-f^{\prime }\theta \right)\right]=0$$

(11)

$$\left(1+{\epsilon }_2\right){\varphi }^{″}+{\epsilon }_2{\varphi }^{\prime }+\frac{N_t}{N_b}{\theta }^{″}+Sc\left(f{\varphi }^{\prime }-f^{\prime }\varphi \right)=0$$

(12)

$$\left.\begin{array}{l}f=s,f^{\prime }=1,{\theta }^{\prime }=-\gamma \left(1-\theta \right),N_b{\varphi }^{\prime }+N_t{\theta }^{\prime }=0 at \eta =0\\ f^{\prime }\to A,\theta \to 0,{\varphi }^{\prime }\to 0,as \eta \to \infty \end{array}\right]$$

(13)

After the reduction of PDEs into ODEs, there are some physical parameters that appear in the system of ODEs.

$$\left.\begin{array}{l}We=\left({\Gamma }^2{x}^2{c}^3\right)/v,s=v_w/\sqrt{cv},Pr=v/\alpha ,\\ {\theta }_r=\left(T_r-T_{\infty }\right)/\left(T-T_w\right)=\left(-1\right)/\gamma \left(T-T_{\infty }\right),\\ G=G{r}_x/\left(R{e}_x^2\right)=\left(g{\beta }_{T}b\right)/c^2,M=\frac{\sigma B_0^2}{\rho a}\\ N_t=\left(\tau D_T\left(T_w-T_{\infty }\right)\right)/\left(v{T}_{\infty }\right),N_b=\left(\tau D_T\left(C_w-C_{\infty }\right)\right)/v,\\ N_t=\left(\tau D_T\left(T_w-T_{\infty }\right)\right)/\left(v{T}_{\infty }\right),N_b=\left(\tau D_T\left(C_w-C_{\infty }\right)\right)/v,\\ A=a/c,G{r}_x=\left(g{\beta }_T\left(T_w-T_{\infty }\right)x^3\right)/v^2,\\ R{e}_x=\frac{ u_w\left(x\right)}{v}, N=\frac{4{\sigma }^{\ast }{T}_{\infty }{}^3}{k^{\ast }{k}_{\infty }}\end{array}\right\}$$

(14)

4. Methodology

The above-obtained ODEs can be handled numerically with the assistance of the Lobatto IIIA MATLAB built-in scheme. In this scheme, the nonlinear model is converted into ODEs with the help of suitable transformations. The dimensionless ODEs are reduced into the first order by taking suitable variables and solving these first-order ODEs with the help of the Lobatto IIIA scheme. The detailed flow chart of the Lobatto IIIA scheme is given below. In the first step the nonlinear PDEs comprise distinguished effects such as the stagnation point, varying viscosity, temperature, changing conductivity, the Buongiorno model comprising nanoparticles, nonlinear based warmth radiation, variation in molecular diffusivity, zero mass flux, and temperature convection. These modeled PDEs are renovated into dimensionless ODEs by considering suitable self-transformations. The dimensionless PDEs are transformed into first-order ODEs by adopting suitable variables as shown in the equations below. These first-order ODEs are handled numerically with the utilization of the Lobatto IIIA MATLAB built-in scheme. The graphical analysis in terms of figures and tables was carried out in the case of various dimensionless parameters that appeared during the numerical simulation of the problem against velocity, temperature, concentration, skin friction coefficient, and Nusselt numbers. The conversion of modeled PDEs regarding mass, momentum, temperature, and concentration into dimensionless ODEs was performed by the procedure mentioned below.

$$f=w_1,$$

(15)

$$f^{\prime }=w_2,$$

(16)

$$f^{″}=w_3,$$

(17)

$$\begin{gathered} f^{‴}=\left(\frac{{\theta }_r-w_4}{{\theta }_r}\right) \left(w_2{}^2-w_1{w}_3+\mathrm{G}{w}_4+A^2+1M^2{\sin }^2\left(\omega \right)w_2\right)+w_3\left(\frac{w_5}{{\theta }_r-w_4}\right) \\ \hspace{1em}\hspace{1em}+{\left(1+{\left(We{w}_3\right)}^n\right)}^{-2} \left(1-\left(n-1\right){\left(We{w}_3\right)}^n\right) \end{gathered}$$

(18)

$${\theta }^{\prime }=w_5,$$

(19)

$${\theta }^{″}=\frac{-\left({\epsilon }_1{w}_5{}^2+\mathrm{Pr}\left(w_1{w}_5+w_2{w}_4\right)+\mathrm{Pr}(N_b{w}_5{w}_7+N_t{w}_5{}^2\right))}{\left(\left(1+{\epsilon }_1\right)+\frac{4}{3}N{\left(1+\left({\theta }_f-1\right)w_4\right)}^3\right)},$$

(20)

$${\varphi }^{\prime }=w_7,$$

(21)

$${\varphi }^{″}=\frac{Sc\left(w_2{w}_7-Sc{w}_1{w}_6\right)-\left(\frac{Nt}{Nb}\right)w_6{}^{\prime }-{\epsilon }_2{w}_7}{1+{\epsilon }_2},$$

(22)

$$\left.\begin{array}{l}\eta =0 :w_1\left(\eta \right)=s, w_2=1, w_5\left(\eta \right)=-\gamma \left(1-w_4\right), N_b{w}_7+N_t{w}_5=0, \hfill \\ \eta \to \infty : w_2\left(\eta \right)\to A, w_4\left(\eta \right)\to 0, w_6\left(\eta \right)\to 0.\hfill \end{array}\right\}$$

(23)

The Lobatto IIIA scheme [35,36] is adopted here to fetch the numerical solution of an assumed problem and its general procedure is explained by a chart in Figure 2.

5. Comprehensive Debate Based on Numerical Results

This segment has been launched to discuss the impact of several physical factors on the rapidity, temperature, and concentricity outlines. The engineering amounts of attention such as the frictional force and Nusselt quantity were calculated and portrayed in the form of figures and tables.

Figure 3 is constructed to study the effect of unsteadiness parameter $A$ on the rapidity field in the absence (presence) of an inclination angle. It is quite evident that amplification in the unsteadiness parameter brings about an increment change in the fluid velocity. The inclination angle magnifies due to magnification in $A$, which escalates the fluid flow and the velocity field. The effect of the Grashof number $G$ on the velocity field in the absence (presence) of the inclined angle $\omega$ is highlighted in Figure 4. The Grashof number is the ratio of buoyancy forces to the viscous forces. A positive change in $G$ intensifies the buoyancy forces instead of viscous forces. The density of the fluid diminishes owing to an enrichment in $G$, which lessens the fluid viscosity and magnifies the fluid velocity. The velocity of fluid diminishes by the virtue of magnification in the power-law index $n$ as shown in Figure 5. The parameter n decides how viscous the fluid is. Fluid behavior in the case of $n<1$ is shear thinning, $n=1$ is Newtonian, and $n>1$ is shear thickening. The fluid becomes much dense, and viscosity amplifies due to an increase in n which brings about an abatement in $f^{\prime }\left(\eta \right)$. An incremental change in electrical conductivity allows more current to flow through the fluid which creates a resistive force termed Lorentz force, which depreciate the fluid motion and furthermore diminish $f^{\prime }\left(\eta \right)$ as shown in Figure 6. Figure 7 reveals the influence of the Weissenberg number $We$ on $f^{\prime }\left(\eta \right)$. The parameter We represents the relaxation time, the time in which the fluid deforms to regain its original shape. During that time the viscosity of the fluid increases, which depreciates the fluid velocity. The Weissenberg number is the ratio of elastic forces to viscous forces. The fluid becomes more viscous by escalating $We$ which lessens the fluid motion and $f^{\prime }\left(\eta \right).$ The heat transfer coefficient $\gamma$ is determined by the transfer of heat from one place to another and the heat transfer rate amplifies on the behalf of magnification in $\gamma$ as shown in Figure 8. It is observed that the rate of convective heat transfer is proportional to the temperature difference. A positive variation in $\gamma$ provides substantial heat which is absorbed by the system and amplifies the temperature field. Radiation is one of the modes of heat transfer, with the other two being conduction and convection. Nonlinear thermal radiation is employed where the elevated temperature change is required and has immense utilization in an industry such as polymer fabrication, combusting apparatuses, satellites, etc. Thermal radiation is directly related to the temperature difference. The temperature of the fluid escalates because of a positive change $N$ which enhances the temperature field $\theta \left(\eta \right)$ as shown in Figure 9. Figure 10 investigates the impact of the Prandtl number on $\theta \left(\eta \right)$. The Prandtl number is a ratio of momentum diffusivity to thermal diffusivity. It is observed that the thermal diffusivity of nanoparticles lessens because of magnification of the Prandtl number $Pr$, which depreciates the temperature of the fluid and heat transfer rate. From Figure 11 it is observed that the temperature field amplifies owing to amplification in ${\theta }_w$. The parameter ${\theta }_w$ is an important part of nonlinear thermal radiation along with $N$. It is observed that the nonlinear thermal radiation term ${\left(1+\left({\theta }_w-1\right)\theta \right)}^3$ escalates by increasing ${\theta }_w$, which augments the energy equation and temperature field and the overall heat transfer rate of the fluid. Figure 12 displays the effect of Brownian diffusion, $N_b$, on the concentration field $\varphi \left(\eta \right).$ Brownian diffusion is one of the important parameters of the Buongiorno nanofluid model. Brownian diffusion is inversely related to the concentration phenomenon. Incremental change in the Brownian motion parameter$N_b$ lessens the concentration phenomenon, which diminishes the concentration field $\varphi \left(\eta \right).$ Figure 13 is designed to study the impact of thermal conductivity ${\epsilon }_1$ on $\varphi \left(\eta \right).$In fluids, thermal conductivity takes place due to intermolecular collision. Molecules collide more randomly because of magnification in ${\epsilon }_1$. The molecules shift kinetic energy, which amplifies the fluid temperature and diminishes the concentration of the fluid and $\varphi \left(\eta \right).$ The thermophoresis parameter $N_t$ is inversely related to $\varphi \left(\eta \right).$ The concentration boundary layer thickness expands owing to an increment in $N_t$ (Figure 14). The concentration of the fluid decreases when the thermophoretic parameter is increased. The impact of raising the thermophoretic parameter is restricted to increasing the concentration profile wall slope. The Schmidt number $Sc$ plays the same role in mass transfer rate as $Pr$ in the heat transfer rate. The Schmidt number is a ratio of mass diffusivity to mass diffusivity. Diffusivity is related to the concentration and Schmidt $Sc$ is inversely related to the diffusion phenomenon. A positive variation in Sc diminishes the diffusion phenomenon, which lessens the concentration and moreover $\varphi \left(\eta \right)$(Figure 15).

6. Validity of This Study

Table 1. Validness of the current scheme.

$\mathit{A}$	Ref. [37]	Ref. [38]	Ref. [39]	Present
0.01	−0.9963	−0.998024	−0.99803	−0.99803
0.02	−0.993	−0.995783	−0.99578	−0.99578
0.05	−0.983	−0.98758	−0.98758	−0.98759
0.1	−0.9603	−0.969386	−0.96938	−0.96938
0.2	−0.908	−0.918107	−0.91811	−0.91811
0.5	−0.6605	−0.66726	−0.66726	−0.66726
1	0.000001	0.000001	0.000001	0.000001
2	2.0181	2.01767	2.01762	2.01761
3	4.72000	4.72964	4.72967	4.72969
Standard Deviation	3.496456	3.496479	3.496489	3.496507

is designed to present the comparative analysis of the obtained results with Hayat et al. [37]; an examination which probed the unsteadiness flowing of Powell-Eyring liquid via an angled stretchable plate was investigated in this paper by employing the homotopy analysis technique in the existence of thermal radiation. Additionally, we compared our results with a non-uniformly heat-generating study by Waqas et al. [38], which elaborated on the unsteadiness magnetic hydrodynamical (MHD) stagnating point, flowing via a porousness shrinkable plate of Casson fluid, flowing with radiative fluxing transference by the shooting method. We also compared our results with Raghunath et al. [39], an investigation which discussed the Soret effects on unsteady magnetohydrodynamics flowing across a semi-infinite vertical penetrable moveable plate with thermal radiative, heat absorptive, and homogeneous chemically reactive samples exposed to changing suction by the Runge-Kutta 4th-order computational technique, in limiting cases by taking variation in unsteady parameter $A$. From the table, it is observed that the obtained computational outcomes for the case of the present model with the utilization of the current numerical technique are in good agreement with the available literature.

Figure 16 displays the graphical comparison analysis of the obtained results with the already-available literature. The graphical comparison analysis tells us that the obtained results are quite authentic and satisfactory.

Figure 17 investigates the effect of $We$ and ${\theta }_r$ on the frictional force coefficient. The surface drag coefficient decreases by increasing $We$ and ${\theta }_r.$ The viscosity of fluid increases by rising $We$ and the fluid becomes more viscous by increasing ${\theta }_r.$ As a result, the skin friction coefficient diminishes. The effect of Grashof number $G$ and power-law index on the surface drag coefficient is highlighted in Figure 18. The buoyancy forces dominate the viscous fluid by increasing $G$, which amplifies the skin friction coefficient, and the density of fluid lessens by improving the power-law index, which strengthens the surface drag coefficient phenomenon.

Figure 19 investigated the influence of ${\theta }_w$ and ${\theta }_r$ on the Nusselt number. A positive change in ${\theta }_w$ amplifies the nonlinear thermal radiation phenomenon which amplifies the heat transmission rate. The heat transmission rate diminishes by enhancing ${\theta }_r.$

The impact of both $N_t$ and $N_t$ on the heat transfer rate is displayed n Figure 20. Both $N_t$ and $N_b$ are important parameters of the Buongiorno nanofluid model. It is well established that the addition of nanoparticles in the base fluid amplifies the heat transmission phenomenon. That is why a variation in $N_t$ and $N_b$ amplifies the Nusselt number.

A positive variation in $N$ and $Pr$ depreciates the Nusselt number as shown in Figure 21. A continuous supply of nonlinear thermal radiation excites the conduction process among the molecules of the fluid. Molecules of the fluid collide more randomly. The Nusselt number is a ratio of heat transfer by convection to heat transfer by conductivity. The Nusselt number is inversely related to the conduction phenomenon. So, an increment in $N$ depreciates the heat transfer Nusselt number. Thermal diffusivity lessens by augmenting $Pr$ which diminishes the heat transfer rate.

Figure 22 investigates the impact of thermal conductivity ${\epsilon }_1$ and molecular diffusivity ${\epsilon }_2$ on the Nusselt number. Diffusivity is related to concentration. A positive variation in diffusivity increases the concentration which amplifies the concentration and diminishes the liquid temperature and heat transmission rate. A variation in thermal conductivity magnifies the temperature within the liquid and depreciates the heat transfer rate phenomenon.

Table 2. Tabulation of skin friction Cross fluid.

$\mathit{W}\mathit{e}$	${\mathit{\theta }}_{\mathit{r}}$	$\mathit{G}$	$\mathit{N}$	Skin Friction
1.1				0.747311
1.3				0.675125
1.5				0.531447
	0.1			0.511412
	0.3			0.340778
	0.5			0.293925
		0.5		0.191472
		0.8		0.193867
		1		0.197345
			0.1	0.318987
			0.4	0.383099
			0.6	0.421181

is designed to reflect the impact of various dimensionless parameters that appear during the numerical simulation of the problem on the frictional force coefficient. It is remarked from the table that magnification in $We$, ${\theta }_r$ decreases the surface drag coefficient but skin friction amplifies in the case of $G$ and $N$.

The trial case of Nusselt quantity for the status of various dimensionless parameters is highlighted in

Table 3. Tabulation of Nusselt number Cross fluid.

${\mathit{\theta }}_{\mathit{w}}$	${\mathit{\theta }}_{\mathit{r}}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{N}}_{\mathit{b}}$	$\mathit{N}$	$\mathit{P}\mathit{r}$	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	Nusselt Number
0.1	0.5	0.3	0.3	0.1	0.7	0.1	0.2	0.189769
0.4								0.1958574
0.7								0.1967127
	0.8							0.1955081
	1.1							0.1954922
	1.4							0.1954817
		0.7						0.1955544
		0.9						0.1955731
		1.4						0.1956091
			0.2					0.1965355
			0.4					0.1985558
			0.6					0.1995958
				0.2				0.3642232
				0.3				0.3460589
				0.4				0.3278681
					1.0			0.1959707
					1.1			0.1930041
					1.2			0.1956082
						0.3		0.1954585
						0.4		0.1955415
						0.5		0.1957222
							0.5	0.1955347
							0.6	0.1955344
							0.7	0.1955346

. It is noted that a positive change in ${\theta }_w$, $N_t$, $N_b$ and ${\epsilon }_1$ provides magnification in the heat transmission rate. The Nusselt quantity diminishes for the case of the rest of the parameters, ${\theta }_r$, $N$, $Pr$ and${\epsilon }_2$.

7. Conclusions

Two-dimensional Cross fluid past a stretchable plate under the effect of Buongiorno nanofluid has been considered. The nonlinear PDEs with the inclusion of various effects were converted into ODEs and tackled numerically with the utilization of the Lobatto IIIA scheme. Some conclusive remarks from the present study are mentioned below.

• An incremental change in variable temperature ${\epsilon }_1$ and molecular diffusivities ${\epsilon }_2$ diminishes the heat transfer rate.
• An amplification in unsteady parameter $A$ improves the velocity field.
• The fluid velocity diminishes by escalating the Weissenberg number $We$ and power-law index $n$.
• A positive variation in the Grashoff number $G$ and thermal radiation $N$ amplifies the temperature field.
• The temperature of the fluid escalates by improving $N$, ${\theta }_w$ and ${\epsilon }_1.$
• Both $N$ and $Pr$ deteriorate the Nusselt number phenomenon.
• The dimensionless parameters $G$ and $n$ magnify the surface drag coefficient.

8. Future Directions

This work can be extended to a hybridity nanofluid model with an inclined magnetic field and heart attack diseases can be controlled by nanoparticles with a magnetic effect. The adopted method of the paper can be implemented in the research articles reported in [40,41,42,43,44,45,46,47,48,49,50,51,52,53] to quantify more physically reliable and numerically stable solutions and to present them as linear models with some criteria for the assessment of the R-Squared, standard error, and $p$-level.